where \(\alpha =0.04\) , \(k=0.3\) , \(n_{p}=\frac{9}{10 \sqrt{2 \pi }}\) , \(n_{b}=\frac{2}{10 \sqrt{2 \pi }}\) , \(u=4.5\) , \(v_{t}=0.5\) . The weight function \(w(v) = \exp (-\frac{v^2}{3})\)  is chosen. We compare the performance of the three low rank methods including the non-conservative method in [1]}, the conservative method in [2]}, and the proposed LoMaC low rank method. In the simulations, we set the mesh size as \(N_x\times N_v=128\times 256\)  and the truncation threshold as \(\varepsilon =10^{-4}\) . The results are plotted in Figure REF . It is observed that all three methods generate numerical solutions with consistent electric energy evolution histories and comparable numerical ranks. Furthermore, it is found that
the non-conservative method can preserve the total mass, total momentum, and total energy up to the truncation threshold \(\varepsilon =10^{-4}\) , the conservative method can conserve the total mass and total momentum on the scale of \(10^{-12}\)  but not the total energy, and the proposed LoMaC low rank method can conserve the total mass, total momentum, and total energy on the scale of \(10^{-14}\) . In Figure REF , we report the contour plots of the solutions by the three methods. Meanwhile, we notice that results by the conservative method and the proposed method are more consistent, which is partly because of their excellent conservation properties.
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